FAKULTAT II

MATHEMATIK UND

NATURWISSENSCHAFTEN

Institut fur Mathematik

BENCHMARKS FORSTRICTLY

FUNDAMENTAL CYCLE BASES

by

CHRISTIAN L IEBCHEN, GREGORWUNSCH,EKKEHARD K OHLER, ALEXANDER REICH,

AND ROMEO RIZZI

No. 003/2007

Benchmarks for Strictly Fundamental CycleBases

Christian Liebchen, Gregor Wunsch,Ekkehard Kohler, Alexander Reich,

and Romeo Rizzi

5th February 2007

Abstract

In the MINIMUM STRICTLY FUNDAMENTAL CYCLE BASIS (MSFCB) prob-lem one is looking for a spanning tree such that the sum of the lengths of its inducedfundamental circuits is minimum.

We identify square planar grid graphs as being very challenging testbeds forthe MSFCB. The best lower and upper bounds for this problem are due to Alon,Karp, Peleg, and West (1995) and to Amaldi et al. (2004).

We improve significantly their bounds, both empirically andasymptotically.Ideally, these new benchmarks will serve as a reference for the performance of anynew heuristic for the MSFCB problem which will be designed only in the future.

1 IntroductionConsider the following problem. Given theN×Nsquare planar grid graphGN,N. Find a spanningtreeT such that the sum of the lengths of its in-duced fundamental circuits is as small as possi-ble. In Figure 1 we provide a very good solutionfor G8,8. Is this optimal?

At first sight, this might appear being a kindof “toy problem.” Indeed, at the occasion ofits annual web-based Christmas quiz (www.mathe-kalender.de), on December 18, 2006 the

Figure 1: A very good SFCB ofG8,8. Itcosts 266. Can you give a cheaper one?

DFG Research Center MATHEON essentially asked the above question to more than9000 registered users (pupils, teachers, scientists, and others). Typically, each dayabout 1500 users post their answers, and more than 60% of these answers are correct.In contrast, on Dec. 18, less than everyeighthanswer has been correct—a first indicatorfor the trickiness of this particular problem.

The fundamental circuits with respect to some spanning treein a general graphform a strictly fundamental cycle basis, where we refer to Section 2 for any formal def-inition. We refer to the problem of finding a spanning tree whose fundamental circuits

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sum to a minimum value as the MINIMUM STRICTLY FUNDAMENTAL CYCLE BA-SIS (MSFCB) Problem. As a generalization, in the MINIMUM CYCLE BASIS (MCB)Problem one seeks for a general cycle basis of minimum length.

Applications. The MCB problem has many applications. These include biology andchemistry ([11]), traffic light planning ([15]), periodic railway timetabling ([17]), andelectrical engineering ([6]). Typically, cycle bases are computed as a kind of prepro-cessing. Then, throughout the actual computations one ensures that a certain problem-specific property is true for the elements of the cycle basis in the graph of interest. Bythis, one can conclude that this property is actually true for anycycle in the graph, rightas it is required by the practical application. In many cases, the shorter the used cyclebasis, the shorter the time for the actual computations.

For some of these applications, due to structural reasons not all cycle bases areof use (e.g. traffic light planning and periodic railway timetabling), but strictly funda-mental cycle bases—being the most specialized ones—alwaysare. In other applica-tions, such as electrical engineering, it is at least much more favorable to use strictlyfundamental cycle bases, because of the numerical stability of the subsequent calcula-tions ([3]). The practical relevance of the MSFCB problem isalso reflected by numer-ous computational studies by different groups working in combinatorial optimization([2, 7, 8, 9, 12, 16, 20]). We shortly overview these works andtheir findings.

Theory. As early as 1982, Deo et al. ([7]) proved the MSFCB problem to be NP-hard for general unweighted graphs. Yet, the many applications require solutions tobe generated anyway. Hence, many heuristics were proposed and tested. However, fornone of these heuristics neither any non-trivial approximation ratio nor any non-trivialbound on the absolute length of the resulting bases was proposed. The only statementinto that direction is that Deo et al. ([7]) conjecture MSFCBs of unweighted graphs tohave lengthO(n2).

The design of most of these heuristics has been led by the following observation:“A BFS produces spanning trees of short diameters. Thus, theBFS method on theaverage generates fundamental cycles of shorter total length (compared to some otherapproaches).” ([7]). In particular, these heuristics makelocal decisions that are mainlybased on the degrees of the vertices, either inG or in some residual graph.

But also there can be applied totally different techniques.Actually, Elkin et al. ([10])consider some average-stretch tree spanner problem. Profiting from the Unified Nota-tion for Tree Spanner problems (UNTS, [19]) one can easily see that in the case of un-weighted graphs their results apply immediately to the MSFCB problem. In particular,their recursive algorithm computes a SFCB of asymptotic lengthO(m· log2nlog logn).Let us emphasize that this is the first non-trivial theoretical guarantee on the quality ofa solution to the SFCB problem, and it is obtained by a recursive approach. Moreover,for graphs with|E| ∈ O(|V|2−ε), this result proves Deo’s conjecture.

Why Planar Grids? In the absence of theoretical bounds for the many degree-basedheuristics, their authors used empirical calculations to assess their quality. But to com-pare different heuristics empirically, these must be run onthe very same input graphs.

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But what aregoodsuch testbeds?Liberti et al. ([16]) consider square planar grid graphs being “the most difficult

testbeds for the MSFCB problem, both for heuristic and exactmethods, due to thehuge quantity of configurations having the same SFCB cost.” In fact, also from a the-oretical perspective this can be motivated in three ways. First, these graphs are almostregular because more thann− 4

√n vertices have degree four—a nightmare for any

degree-based heuristic. Second, within a fixed distance, the subgraphs around almosteach vertex are isomorphic. Hence, any heuristic that basesits decisions on local con-figurations risks to perform poorly. Third, ifG was a tree, then in the MSFCB problemno decisions are to be made and the problem clearly becomes trivial. An appropriatemeasure for the tree-alikeness of a graph is its tree-width ([4]). And with respect tothat measure, grid graphs—havingΘ(n) edges and tree-width

√n—are prominent ex-

amples of being far away from being a tree ([22]). Thus the MSFCB problem is likelyto keep its hardness. In addition, in several applications relevant instances are planargraphs, sometimes even grids (e.g. electrical engineering, traffic light scheduling).

Focusing on grid graphs could appear narrow. But it is commonly believed thatthese hold the key to better algorithms. Indeed, for square planar grid graphs Alon,Karp, Peleg, and West ([1]) design spanning trees of length4

3nlog2n+ O(n) ([14]).They prove these trees to be asymptotically optimal. Moreover, they conjecture theirtrees are “essentially optimal.” This asymptotical upper bound lets us demonstrate howdegree-based heuristics may fail. The degree-basedC -order heuristic can be imple-mented to compute “Machete”-trees (cf. [5], and Figure 1 foran example). These treesdo not only minimize the diameter of trees in grids, but also the maximum stretch. Atfirst sight these two parameters of a tree could appear being tightly related to its asso-ciated SFCB cost. It is again the UNTS ([19]) which makes it transparent that even forunweighted graphs no two of these three measures do actuallycoincide. As Machete-trees yield MSFCB objective values ofΘ(n

32 ), on grid graphs degree-based heuristics

risk to fail drastically. This underlines grids being a relevant testbed.As a matter of fact, Liberti et al. ([16]) select grid graphs as one of their testbeds.

On the 50×50 grid they observe that their newC -order heuristic attains an objectivevalue of 46452, compared to 48254 of the NT heuristic. Unfortunately, from this iso-lated comparison it has to remain unclear whether these are good objective values at all.In fact, Amaldi et al. ([2]) also consider grid graphs in their computations. And they re-port a solution of objective value 23026, that was obtained by local search techniques.This motivates the need for clear benchmark values for the MSFCB problem for theparticularly challenging case of planar grid graphs—also for the future evaluation ofnew heuristics.

Of course, relevant benchmarks also include dual bounds. Since general cycle basesare a superset of strictly fundamental cycle bases, the value of an MCB clearly servesas a lower bound for the value of an MSFCB. On grid graphs, thisyields a lower boundof 4 · (√n−1)2. But exploiting the particular structure of grid graphs onecan achieveasymptotically better lower bounds for the MSFCB problem. The first was given in [1]and it has valueln2

2048nlog2n−O(n).

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Contribution. The above discussion motivates a need for a collection of benchmarkvalues for the MSFCB problem on square planar grid graphs. Weprovide two newfamilies of lower bounds and two new families of upper bounds.

In Section 3 we sketch a proof of Kohler et al. ([14]) on how a new approach raisesthe asymptotical lower bound by Alon, Karp, Peleg, and West ([1]) to 1

12nlog2n−O(n), i.e. by a factor of more than 245. In addition, we identify 6n−20

√n+22 as a

new lower bound. ForN ∈ {3, . . . ,61} this constitutes the best known lower bound. Itis a fact thatall the primal solutions (upper bounds) that so far have been proposed inthe literature are grids which fall within this range.

Finally, in Section 4 we introduce a new scheme for constructing very short strictlyfundamental cycle bases—both empirically and asymptotically. We prove an upperbound on the length of their SFCB of 0.97nlog2n+ O(n), hereby improving the ob-jective value 4

3nlog2n+ O(n) of the spanning trees due to Alon, Karp, Peleg, andWest ([1]), which they assumed being essentially optimal. In our experiments we alsocompare their lengths to spanning trees that were obtained by using local search tech-niques ([2]). It turns out that our new trees improve the bestsolutions known so far forall N ≥ 20. Interestingly, forN = 10,15, . . . ,55 they even constitute local optima withrespect to the 2-neighborhood.

2 Preliminaries

We consider cycle bases of a 2-connected simple undirected unweighted1 graphG =(V,E). Definen = |V|, m= |E|, andν = m−n+1, whereν is thecyclomatic numberof G. Let C be a circuit (cf. [23, Ch. 3]) inG and denote byγC its {0,1}-incidencevector. Thecycle spaceC of G is the following vector subspace over GF(2),

C := span({γC |C circuit in G}) .

A cycle basis Bof G is a set ofν circuits ofG whose incidence vectors are a basis ofC .ThelengthΦ(B) of a cycle basis of an unweighted graph is defined asΦ(B) = ∑C∈B |C|.A minimum cycle basis(MCB) of a graphG is a cycle basis ofG of minimum length.

A set of circuits{C1, . . . ,Cν} such that

Ci \ (C1∪·· ·∪Ci−1) 6= /0, ∀i = 2, . . . ,ν

is clearly a cycle basis. We call such a basisweakly fundamental. Notice that thesewere already considered by Whitney ([24]) in 1935.

Let T be some spanning tree ofG. Depending on the context, we either regardTas a subgraph ofG or as a set of edgesT ⊂ E. Fore∈ E \T, we denote byCT(e)—orCe for short—thefundamental circuitthate induces with respect toT, i.e. the uniquecircuit in T∪{e}. ToT there are associatedν fundamental circuits. These form a cyclebasis which is calledstrictly fundamental. Here, we may writeΦ(T) instead ofΦ(B).A minimum strictly fundamental cycle basis(MSFCB) has minimum length among the

1Of course, minimum cycle basis problems are also investigated for weighted graphs. But as we aimto contribute to the particularly challenging case of planar unweighted grid graphs, we omit edge weightsthroughout our presentation.

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set of strictly fundamental cycle bases. In the context of local search, for an arbitraryspanning treeT we define its 2-neighborhoodas the set of spanning treesT ′ such that|T ∩T ′| ≥ |T|−2.

In general, strictly fundamental cycle bases are a proper subset of weakly funda-mental cycle bases, which in turn are a proper subset of general cycle bases of undi-rected graphs. Moreover, in general no two of the three corresponding minimizationproblems coincide ([18]).

With N∈N, the planargrid graph GN,N is the graph onV = {1, . . . ,N}×{1, . . . ,N}with

E = {{(i, j),(i′, j ′)} : |i− i′|+ | j − j ′| = 1} = {{u,v} : ||u−v||1 = 1}.

In a graphical representation, e.g. in an embedding intoZ2, the first index of a vertexrepresents itsx-coordinate, the second index itsy-coordinate. The graphGN,N hasn =N2 vertices and containsm= 2·N ·(N−1) edges. Its cyclomatic numberν is (N−1)2.

We denote the dual of an embedded planar graphG by G∗. The graph(GN,N)∗

is again the graph of a square(N−1)× (N−1) grid plus a further vertexF∞, whichcorresponds to the outer face of the initial embedded planargraph. Recall from [23,Ch. 3] that the edge set ofG can be identified with the edge set ofG∗.

Now, consider a spanning treeT of GN,N and its dual counterpart, that we denote byT∗. In fact,T∗ can be understood as the complement ofT, as it contains the counterpartin G∗ of each edge inE(GN,N)\T. The graphT∗ is a spanning tree ofG∗, although itis not necessarily connected when restricted toG∗ \{F∞}.

3 New Lower Bounds

Trivial lower bounds for the MSFCB problem are the length of aminimum weaklyfundamental cycle basis, or even of an MCB ([24, 18]). Whereas the former is in gen-eral APX-hard to find ([21]), for the latter there are known polynomial-time algorithms(e.g. [13]). However, as a consequence of a result due to Alonet al. ([1]) one canconclude that these lower bounds risk to miss the optimum value of the MSFCB prob-lem by a logarithmic-factor. And this is in particular the case for square planar gridgraphsGN,N. Here, the trivial lower bound is only 4· (√n−1)2 whereas Alon et al.provedΦ(T) ≥ lnn

2048n−O(n) for all spanning treesT.An alternative way to obtain lower bounds is to consider MIP formulations of the

MSFCB problem (Liberti et al., [16]). Later, in [2] there wasidentified an improvedMIP formulation, which turns out to be more efficient in theirempirical computations.With this MIP formulation, we effected some spot tests on square planar grid graphs.Unfortunately, we had to observe that it is alreadyveryhard for standard MIP solvers—such as CPLEXc© 10.1—to get beyond the trivial lower bound. This is why, in Sect. 3.1we identified several classes of valid inequalities for these MIPs, some of them beingdefined even for general graphs. Indeed, these helped MIP solvers to detect better lowerbounds.

Yet, in Corollary 6 we identify 6n−20√

n+ 22 being a lower bound for the MS-FCB problem on a square planar grid, using purely combinatorial arguments. And

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then again, it got completely unpractical for the MIP—even in our refined version—toimprove on this lower bound.

Of course, there exists some dimensionN0 such that our lower bound gets domi-nated by the asymptotically better lower bound by Alon et al.([1]). But in the caseof N = 2k +2, k∈ N we also present one further lower bound function with

Φ(T) ≥ 112

nlog2 n−O(n), (1)

which is due to Kohler et al. ([14]). Clearly, it dominates Alon et al.’s lower bound.And it illustrates the predominance of the 6n−20

√n+22 bound over the trivial lower

bound: The function that interpolates the asymptotically best lower bound (1) intersectswith the trivial lower bound as early asN1 ≈ 8.1. But 6n−20

√n+ 22 intersects (1)

only atN2 ≈ 61.6, i.e. in the case of more than 7300 edges.

3.1 A MIP Formulation

In this section we qoute the MIP formulation by Amaldi et al. ([2]) for the MSFCBproblem on general graphs. Moreover, we make it more efficient by identifying thefirst classes of valid inequalities. In particular, for planar grids two of these classes areeven able to cut offanyof the huge number of optimum solutions of the LP relaxation,hereby improving the lower bound of the root node in the Branch-and-Bound tree.

Let G= (V,E) be a 2-connected graph with non-negative costswe on an edgee∈E.To ensure a spanning treeT to be computed, we resort on the following characteriza-tion: |T| = |V|−1, andT is connected. We exploit the fact thatT is connected, if andonly if for each non-tree edgee= {i, j} ∈E\T there exists a path inT betweeni and j.

Amaldi et al. introduce a binary variablezi j for each edge{i, j} ∈ E, wherezi j = 1iff e∈ T. Of course, the correct cardinality ofT is easy to state. Then, they are going toensure the non-tree edge connectivity by introducing non-negative variablesx, whichare chosen to be well-suited to state the objective functionof the MSFCB problem. Thevariablesx can be understood as a multi-commodity flow inG, only using edges ofT.For each edgee= {k, ℓ} ∈ E, its endpoints are regarded as source and sink of a com-modity for which one unit of flow is to be sent. As this flow may only use tree edges,the commodities inE \T guaranteeT being connected. To state flow conservation,a directed graphD is derived fromG, whose arc set consists of a pair of antiparallelarcs for each edge inE(G), and the costwi j of an arc(i, j) is set equal to the cost ofthe edge{i, j}. Then, the variablexkℓ

i j encodes the directed flow through arc(i, j) forcommodity{k, ℓ}.

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min ∑{k,ℓ}∈E

∑(i, j)∈A

wi j xkℓi j + ∑

{i, j}∈E

(1−2zi j ) wi j (2a)

∑j∈δ(k)

(xkℓk j −xkℓ

jk) = 1 ∀{k, ℓ} ∈ E (2b)

∑j∈δ(i)

(xkℓi j −xkℓ

ji ) = 0 ∀{k, ℓ} ∈ E, ∀i ∈V \{k, ℓ} (2c)

xkℓi j ≤ zi j ∀{k, ℓ} ∈ E, ∀{i, j} ∈ E (2d)

xkℓji ≤ zi j ∀{k, ℓ} ∈ E, ∀{i, j} ∈ E (2e)

∑{i, j}∈E

zi j = n−1 (2f)

xkℓi j ≥ 0 ∀{k, ℓ} ∈ E, ∀(i, j) ∈ A (2g)

zi j ∈ {0,1} ∀{i, j} ∈ E. (2h)

In any integer feasible solution, within the first term of theobjective function (2a) wefind that

∑a∈A

waxea ≥ dT(e), for all e∈ E, (3)

where in any optimum solution equality holds.Although the MIP formulation (2) has been observed to behavebetter than other

formulations ([16]), still there are some major shortcomings. First, the number ofvariables and constraints is large. For instance, there are2 ·m2 x-variables—in otherwordsΘ(N4). Already with this simple observation one might not expect too muchfor the solvability with, say,N ≥ 20. But the second drawback is even worse. TheLP relaxation has several trivial optimum solutions. For instance, takez≡ 1

2 + 12N . This

particular choice admits thex-variables to sum up to 4· (N−1)2, being the optimumvalue of the minimum weakly fundamental cycle basis problemon GN,N. We willprovide another set of optimum solutions of the LP relaxation in Example 2. Of courseone can check this to be the optimum value of the LP relaxationby having a look atthe dual problem. We conclude, adding valid inequalities to(2) will be key for itssolvability.

Thus, in the remainder of this chapter we provide three classes of valid inequalities:two which are valid for general graphs and which are defined either in z-variables orin x-variables, and one class that exploits the particular structure of grid graphs andhereby can combinex- andz-variables.

Lemma 1.Consider the graphG = (V,E) and an arbitrary proper subsetH of E. De-note byV(H) the vertices incident to edges inH. If V(H) ( V, then

∑{i, j}∈(V(H),V(H)){i, j}/∈H, i, j∈V(H)

zi j ≥ |V(H)|− ∑{i, j}∈H

zi j (4)

are valid for every integer feasible solution of the MIP.

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Proof. In an integer feasible solution the edges withzi j = 1 form a spanning treeTof G. Therefore, the right-hand side (RHS) of (4) equals the number of connectedcomponents of the graph with vertex setV(H) and edge setH. As we assumeV \V(H)to be nonempty, each component of(V(H),H) must be reachable from the formervertex set. Hence, to ensure the connectivity ofG via edges for whichzi j = 1, theremust be at least as many such edges, as(V(H),H) has connected components.

On the one hand, we are aware of graphs in which there exist fractional vec-tors(x,z), which are feasible for the LP relaxation and whose objective value is strictlysmaller than that of an integer optimum solution. Hence, these inequalities could ap-pear to be reasonable candidates to add to the LP-relaxationof MIP (2).

On the other hand, it may happen that in an iterative cutting plane generation, onecould only profit from inequalities of that type at rather late iterations.

Example 2.Consider the grid graphGN,N. In Figures 2(a) and 2(b) we sketch twospanning treesT1 andT2 of GN,N. Denote the corresponding solutions of the MIP (2)by (x1,z1) and(x2,z2), respectively.

(a) T1 (b) T2 (c) T3

Figure 2: The Figures 2(a) and 2(b) depict thez vectors of the MIP solutions of twospanning treesT1 andT2 (in bold) for G9,9. Figure 2(c) shows their convex combina-tion z3 = 1

2z1 + 12z2

Now, consider the convex combination of(x1,z1) and(x2,z2) which results in thefollowing fractional vector(x,z) := 1

2(x1,z1) + 12(x2,z2). Clearly, (x,z) is a feasible

solution for the LP relaxation of MIP (2). But observe that there exists a vector(x′,z)which is feasible for the LP, too, but in which

∑{k,ℓ}∈E

∑(i, j)∈A

x′kℓi j =

{

1, if zkℓ = 1 and2, otherwise.

(5)

In particular, the objective value of(x′,z) equals 4· (N−1)2. But this is just the op-timum value of the LP relaxation of MIP (2). Asz is the convex combination of twointeger feasible points, any inequality that does not make use of anyx-component, willnever cut off(x′,z), thus never increasing the LP value.

From the above example we conclude that for planar grids no valid inequality hav-ing non-zero coefficientsonly in z-components whatsoever, will ever be able to cut

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off one particular optimum solution of the LP relaxation, hereby never increasing theoptimum value of any so refined LP.

This is why in the sequel we investigate valid inequalities which are either definedpurely in terms ofx-variables, or as a combination ofx- andz-variables.

Lemma 3.Let G = (V,E) be a 2-connected graph with a spanning treeT and considera simple circuitC in G. Then,

∑e∈C

dT(e) ≥ 2· (|C|−1). (6)

Proof. Let T denote an arbitrary but fixed spanning tree ofG. In the following we willprove the claim by induction over|C\T|. Notice first that|C\T| = 0 would implyC⊆ T, which contradictsT being cycle-free.

Therefore, we select|C\T|= 1 as the inductive base. In this case, the distancesdT(e)in the tree are one for the|C|−1 tree edges, and|C|−1 for the unique non-tree edge.Hence, the claim holds.

In the inductive step, take a circuitC for which |C\T| = k ≥ 2. We will identifytwo circuitsC1 andC2, each with 1≤ |Ci \T| < k, i = 1,2. Then, from (6) being truefor C1 andC2 we argue that the claim is true forC.

Consider two verticesu andv within C which are connected through a pathP⊆ Tsuch thatV(P)∩V(C) = {u,v}, but E(P)∩E(C) = /0. Such a path exists because of|C\T| ≥ 2, otherwiseT would not be connected. Denoting byP1 andP2 the two pathsbetweenu andv defined byC, C1 = P1∪P andC2 = P2∪P are simple circuits inG.

Now, asT is cycle-free, bothP1 andP2 must include at least one non-tree edge,sayeP1 andeP2. OtherwiseT would have contained a cycle. But then,C1 contains atleast one non-tree edge less than the circuitC, because it omitsP2, thuseP2, and thepathP contains only tree-edges. The very same holds forC2.

We may thus apply the inductive assumption toC1 andC2. Summing up (6) forthese two circuits yields

∑e∈C1

dT(e)+ ∑e∈C2

dT(e) ≥ 2· (|C1|+ |C2|−2).

By the construction ofC1 andC2 it holds that|C| = |C1|+ |C2|−2· |C1∩C2|, and thus

∑e∈C

dT(e)+2· ∑e∈C1∩C2

dT(e) ≥ 2· (|C|−1)+4· |C1∩C2|−2. (7)

But sinceC1∩C2 = P⊆ T we have thatdT(e) = 1 for all e∈C1∩C2 and Equation (7)simplifies to

∑e∈C

dT(e) ≥ 2· (|C|−1)+2· |C1∩C2|−2. (8)

Finally, because of|C1∩C2| ≥ 1, and thus 2· |C1∩C2|−2 ≥ 0, Equation (8) implies(6) for the circuitC.

Corollary 4. Let G = (V,E) be a 2-connected graph and consider a simple circuitCin G. Then,

∑e∈C

∑f∈A

xef ≥ 2· (|C|−1) (9)

is a valid inequality for every integer feasible solution ofthe MIP (2).

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Note that the above corollary holds for general graphs. Yet,we are interested mostin investigating its effect on grid graphs.

3.2 Small Grids

Let C be some circuit inGN,N. We denote by diamH(C) thehorizontal diameterof C,i.e. the difference between minimum and maximumx-coordinates inZ2 of verticesin C. Similarly, we denote thevertical diameterof C by diamV(C). In particular,

|C| ≥ 2· (diamH(C)+diamV(C)). (10)

For e 6∈ T, we use diamH(e) := diamH(CT(e)) as a short hand.LetC be a circuit inGN,N and consider its enclosed finite regionR. In C we collect

all the edges inE(GN,N) that are incident with two faces ofGN,N that have emptyintersection withR2\R. In other words,C refers to the edgesinside C.

Proposition 5.Let GM,N be theM×N planar grid,T be a spanning tree in it, andC bea simple circuit inGM,N. Then

∑e∈(C∪C)\T

|CT(e)| ≥ 4· |C\T|+6· |C\T|. (11)

Proof. Using (10) it suffices to establish that

∑e∈(C∪C)\T

2· (diamH(e)+diamV(e)) ≥ 4· |C\T|+6· |C\T|. (12)

We derive a lower bound for∑e∈(C∪C)\T diamH(e) + diamV(e) by defining a func-tion d(e) such that

diamH(e)+diamV(e) ≥ d(e), for all e∈ (C∪C)\T. (13)

We define the functiond(e) as follows. Bye 6∈ T we already know that

diamH(e) ≥ 1 and diamV(e) ≥ 1. (14)

To increased(e) beyond two, consider the spanning treeT∗ in the dual graph(GN,N)∗

that corresponds toE(GN,N) \T. TakeF∞ as the root ofT∗. Consider the two facesof GN,N that are incident withe. We refer to the one with the larger distance fromF∞

in T∗ asF(e).For each edgef ∈ (F(e) \ (C∪{e}), denote byF( f ) 6= F(e) the other face thatf

is incident with. Observe thatF( f ) 6= F∞ because off 6∈C. By the grid structure, eachof these facesF( f ) is in a different direction with respect toF(e), i.e. either north,east, south, or west. Now, iff 6∈ T, we know thatCT(e) also must haveF( f ) in itsenclosed bounded region. This way, such an edgef ∈ (F(e)\ (C∪T ∪{e})) serves asa certificate that any lower bound on either diamH(e) or diamV(e), respectively, can beincremented. In total, we set

diamH(e)+diamV(e) ≥ 2+ |F(e)\ (C∪T ∪{e})| =: d(e), (15)

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which guarantees (13) still to be true.When summing over all edgese∈C∪C, we may rearrange the summation. To this

end, observe that each edgef ∈ C\T has precisely one dual parenteamong(C∪C)\T.Hence, it increments the lower bound on diamH(e)+ diamV(e) for precisely this oneedgee. In other words, each edgef ∈ C\T counts three times: according to (14)it countsd( f ) = 2 for its proper fundamental circuitCT( f ), plus one increment forprecisely its unique parent edgee∈ (C∪C)\T. To summarize,

∑e∈(C∪C)\T

d(e)(15)= 2· |C\T| + 3· |C\T|. (16)

Finally, we conclude that

∑e∈(C∪C)\T

|CT(e)|(10)≥ ∑

e∈(C∪C)\T

2· (diamH(e)+diamV(e))

(13)≥ ∑

e∈(C∪C)\T

2·d(e)(16)≥ 4· |C\T| + 6· |C\T|.

Corollary 6. Let N ≥ 3 andGN,N be theN×N planar grid withn = N2 vertices. Thenfor each spanning treeT ⊂ E

Φ(T) = ∑e∈E\T

|CT(e)| ≥ 6·n−20√

n+22. (17)

Proof. Simply takeC as the circuit that contains precisely the edges that are incidentwith F∞. Because ofE = C∪C we apply Proposition 5 toC. There, we minimizethe RHS in (11) by maximizing|C\T|. Now consider the four vertices which are notincident to any edgeC. In any treeT, these must be incident with one edgeC∩T.As N ≥ 3, we conclude that|C∩T| ≥ 4, thus|C\T| ≤ 4

√n− 8. Finally, a simple

calculation yields (17).

Observe that 6n−20√

n+22≥ 4·n−8√

n+4, for all N ≥ 3 andn = N2.As a special case, consider a spanning treeT in which for each edgee∈ E \T its

distance inT∗ from F∞ is at most two. Then, in particular in (15) equality holds. Inthe end, one can then argue that in (17) we will find equality, too. As forN ∈ {3,4,5}there exist such spanning trees, we conclude that in these dimensions the bound inCorollary (6) is nothing but the optimum value of the MSFCB problem.

3.3 Large Grids

In this section we sketch that the strictly fundamental cycle basisB of any spanningtree T in the squareN×N grid with n = N2 = (2k + 2)2 vertices satisfiesΦ(T) ≥112nlog2n−O(n). Hereby, our direct approach substantially improves the lower boundthat has been obtained by Alon, Karp, Peleg, and West in [1, Thm. 6.6]—by a factor of

11

more than 245.2 Due to space limitations, unfortunately we cannot present any of ournew proofs here. But we refer to [14] for the complete analysis.

In contrast to [1] we decided to tacklethe lower bound problem from the perspec-tive of the planar dual graphG∗. In par-ticular, we exploit that for each spanningtree T, there is a one-to-one correspon-dence between its induced fundamental cir-cuits inG, and its induced fundamental cutsin G∗. More precisely, if an edgee∈ E \Tinduces a circuit inG, then its dual counter-part induces a cut inG∗—and both containthe very same edges.

To detect sufficiently long circuits, ac-cording to Inequality (10) we resort on cir-cuits that have large horizontal and/or verti-cal diameters. To obtain the claimed lowerbound, it even turns out to be sufficient to ei-

Figure 3: The dual treeT∗ of a spanning treeT inG18,18. In our lower bound forΦ(T) we only sumlower bounds on the fundamental circuits that areinduced by the black edges

ther consider only the horizontal diameter diamH(C) or its vertical diameter diamV(C)of a circuitC.

We find circuits having large horizontal or vertical diameter by considering partic-ular faces ofGN,N, or vertices of(GN,N)∗, respectively. The vertices that we use arethe ones that are highlighted in Figure 3. In more detail, we organize these verticesin what we will call levels. We also assign a box to each such vertex. In a dual gridwith (N−1)2 =

(

2k +1)

·(

2k +1)

vertices we establishk different levels of verticesas follows. Thelevel konly contains the unique grid’s center vertex. Its correspondingbox equalsV(GN,N). The four quarters of the grid—which overlap on their borders—become the boxes of levelk−1. By the particular choice ofN, their center vertices arewell-defined, and these constitute the vertices of levelk−1. Recursively, each of thesefour quarters is again subdivided into four new quarters, which become the boxes ofthe next level, and their centers are the corresponding level-vertices. For a vertexu oflevel ℓ = 1, . . . ,k we call the set

{

v : du,v = 2ℓ−1}

theborderof its box Bu.For each such vertexv, we consider the subpathP of T∗ that connects it withF∞.

But we only follow this path until the first edgee that is incident with the border ofBu.Assume w.l.o.g. that the edgee is “north” of v. In the primal grid, in facte is a hor-izontal edge. Then we only consider the subsequence of vertical edgesPV (all beinghorizontal edges inGN,N) of P⊆ T∗ such that each edge is by one closer toe than itspredecessor inPV . In [14] we call this subpath the verticalpseudo-pathof P. Then, fora vertexv of one particular levelℓ, ℓ∈ {1, . . . ,k}, we know that the 2ℓ−1 edges ofPV in-duce fundamental circuits with respect toT of vertical diameters at least 1,2, . . . ,2ℓ−1.

Now, we aim at summing the lower bounds on the diameters of thefundamentalcircuits for each occurrence of an edge on some pseudo-path.It is a simple observationthat only pseudo-paths of different levels could share a, say, horizontal edgeeof GN,N,and thus potentially cause some double-counting (Lem. 3 in [14]). But one can further

2Let us mention that the authors of [1] state explicitly that they were not trying to “optimize constants.”

12

observe that the lower bound on diamV(e) is always larger for the pseudo-path thatwe defined for the vertex with the higher level index. Hence, to prevent us from anydouble-counting, for such an edgeewe count as lower bound on|CT(e)| only the boundon diamV(e) that we identify on the highest level. Doing so yields

Theorem 7([14]). Let GN,N be the planar grid graph withn = N2 = (2k +2)2 vertices.For every spanning treeT of GN,N there holds

Φ(T) ≥ 112

nlog2 n−O(n). (18)

4 New Upper Bounds

Alon et al. ([1]) provided spanning treesTAKPW whose induced strictly fundamental cy-cle bases they showed to be bounded from above by 2nlog2n+ o(nlog2n). An exactcounting ([14]) revealed even that

Φ(TAKPW) ≤ 43

nlog2 n+O(n), (19)

whereN =√

n, andN = 2k for somek∈N. Although Alon, Karp, Peleg, and West ([1])think of their trees as being essentially optimal, we are able to construct trees with anasymptotic coefficient for thenlog2n term being strictly smaller than one. Moreover,we present trees which empirically perform very well already in small dimensions.

Fortunately, we are able to introduce a class of recursivelydefined trees that accom-modatesbothgoals. These spanning trees are the union of spanning trees in rectangu-lar subgraphs ofGN,N, their building blocks. The trees differ in how their rectangularsubgraphs—all respecting some arbitrary but fixed aspect ratio α ≥ 1—partition thefaces ofGN,N. Hence, it remains to specify how to construct a spanning tree subjectto a given parameterα for some gridGM,N havingaspect ratiomax{M

N , NM} ≈ α. This

is done recursively. Assume w.l.o.g. thatM ≥ N. At the top-level of the recursion,we add toTα(GM,N) the edges of the two longer borders ofGM,N (here the horizontalones), plus of one of its two other borders (cf. Figure 4). Forthe recursion, we partition

. . . . . .

...

...

sub-blocksub-sub-block

Figure 4: The shape of a block (left) and with a sketched interior recursively filled withsmaller blocks (right), always keeping the aspect ratio.

the faces ofGM,N into almost equally-sized rectangular subgraphs of aspectratio againbeing close toα; only the faces of one horizontal path in(GM,N)∗, located almost in

13

the middle of its two horizontal borders, are not contained in any of these rectangularsubgraphs.

These trees are related to other families of trees as follows. In GN,N, choosingα ≥ N

2 : 1 there exists a partition of the grid such that we end with Machete-trees ([5],cf. Figure 1). Moreover, an aspect ratio ofα = 1 : 1 yields trees which can be obtainedalternatively by a construction that is much similar to the one forTAKPW.

According to the requirement—asymptotical or empirical quality—we will againsubdivide our presentation into a part considering large grids and into a second partdealing with small grids. For both types of grids we will introduce trees with a blockstructure either having an aspect ratio of approximately 3 :1 or an aspect ratio of 2 : 1,respectively. In addition, the trees differ in how the blocks are actually used to define atree. Whereas on large grids it is sufficient tocoverthe grid with three (almost) equally-sized 3 : 1 blocks, for small dimensions the grids aretiled with many 2 : 1 blocks ofmany different sizes.

4.1 Large Grids

To achieve a good asymptotical upper bound we decided to construct trees out of theabove described blocks with an aspect ratio of 3 : 1. Unfortunately, it turns out to betricky to subdivide or tile a square grid of arbitrary dimension with these particularblocks. Thus, we construct our trees bottom-up like. That means we take an atomicblock of size 6×14 and arrange 32 copies of such a block to a new one having size80×14. This procedure is then iterated providing spanning trees for dimensions

(

1248496

·32k/2 +1531

)

×(

419496

·32k/2 +3031

)

(20)

with k chosen integral and even. Finally, three copies of such a tree can be put onto eachother and cover the entire square grid. Now, a detailed description of the constructionof the tree and a precise analysis of it follows.

Construction of the tree. Whereas in Sec.4 we gave a brief top-down description ofthe tree that we consider we now introduce them bottom-up like, hereby having morecontrol on the dimensions and, thus, by-passing rounding indispositions.

For everyk ∈N+ we construct recur-sively spanning trees as follows. Forkequal to 1 consider the spanning treeT1

as sketched in Figure 5. This tree is de-fined on a 6× 14 grid and it has itsexiton the lower horizontal border. The nexttree, T2, is constructed by arranging 32copies ofT1. First, 16 copies are gluedwith this particular orientation side byside. Second, we mirror the other 16 copiesof T1 horizontally and place them such that

6

14

eT1

Figure 5: The spanning treeT1 out of which all thetreesTk are constructed.T1 has dimension 6×14, orside-lengths 5×13, respectively.

their exits are opposite to the first 16 copies. At last, one vertical edge, which we will

14

call eT2 is added to connect the two so-constructed connected components.The general rule here is to take the left vertical edge as connecting edge for the con-

struction of the treeTk with k even and the upper horizontal edge for the constructionof theTk with k odd. See Figure 5 for an example. By this construction, the treeT2 isof dimension 81×28. In general, the treeTk is constructed out of 32 copies ofTk−1

and an additional connecting edge the very same way.In order to finally state a spanning tree for a square grid and to prepare the

analysis of the trees we introduce four sequences for thex andy length, w.r.t. edges,of Tk in dependence ofk. As T1 is a 6×14 grid tree we havex1 = 5 andy1 = 13. Byconstruction, we get the following sequences taking the parity of k into account:

x2i = 16·x2i−1 y2i = 2·y2i−1 +1 (21)

for treesTk with k = 2i even. For oddk = 2i +1 the treeTk has dimension

x2i+1 = 2·x2i−1 +1 y2i+1 = 16y2i−1. (22)

In the following we will only consider the spanning treesTk for k even. Simplecalculations transform (21) and the start valuesx2 = 80 andy2 = 27, respectively, intothe explicit sequence of the lengths ofTk for evenk:

xk =7831

·32k/2− 1631

yk =419496

·32k/2− 131

. (23)

If we now take a closer look atTk, k even, we see that the ratio of its lengths isalmost 3×1. In fact, the exact ratioxk to yk is always greater than 2.96 and convergesto 1248

419 ≈ 2.978.Hence, if we take three copies ofTk and put them one upon another, then the result-

ing spanning tree, let us denote it byT3k , covers a grid of dimension

(

1248496

·32k/2 +1531

)

×(

1257496

·32k/2 +3031

)

.

We now claim that three times the size ofTk is an upper bound on the size ofT3k

restricted to the square gridG with dimension(

1248496

·32k/2 +1531

)

×(

1248496

·32k/2 +1531

)

.

So, how do we restrictedT3k to a square of the above size? Due to space limitations

we will only give an idea of the procedure. Let us consider theboundary lineL of Gthat, in a sense, cuts through the down-most copyTk of T3

k , cf. Figure 6. ThisTk

consists of several subtreesTk−1, Tk−3, . . . , T1. Thoseodd subtrees can have their exitpointing downwards,↓ Tj , or upwards,denoted by↑ Tj . If for a j = 1,3, . . . ,k−3,k−1the boundary lineL cuts through a subtree↓ Tj we leave this part of tree unchanged andsimply cut away what overhangsL. In the other case where the boundary lineL cutsthrough a subtree↑ Tj we cut away the overhanging parts as well, but—since we looseconnectivity—we add an edge to↑ Tj exactly where formerly the exit had been. If wedo so for all j = 1,3, . . . ,k−3,k−1 we finally come up with a tree with less chordsinducing lower length fundamental cycles than this down-most copyTk of T3

k before.

15

. . . . . .

. . . . . .

f1

L

f2f2f2

Tk−2

↓ Tk−1

↑ Tk−1

16 copies

Figure 6: A schematic illustration of the treeTk for an evenk ∈ N+. Due to theconstruction rulesTk consists of322 = 1024 copies ofTk−2 and different slots, i.e. onemain slot f1, dark-gray, and 32 subslotsf2, light-gray.

Analysis of the tree. So, for everyk∈N+ let us denote withTk the spanning tree ofthe

(

1248496

·32k/2 +1531

)

×(

1248496

·32k/2 +1531

)

grid as described above. We are interested in an upper bound on the strictly fundamen-tal cycle basis induced byTk. As foreshadowed above we have

Φ(Tk) ≤ 3·Φ(Tk). (24)

In the following we develop a recursive formula forΦ(Tk). Because of the tree’s specialconstruction the following recursive formula holds,

Φ(Tk) = 1024·Φ(Tk−2)+ f (Tk), (25)

where f (Tk) denotes the size of the fundamental cycles induced by edges that do not lieentirely within a copy of the smallerTk−2 tree. We call those areas slots. Then,f (Tk)can be canonically subdivided into one main-slot and several sub-slots, cf. Figure 6.Obviously,

f (Tk) = f1(Tk)+32· f2(Tk) (26)

holds. Then, with the help of the sequences defining the lengths of the trees (Equa-tions (21) and (22)) we straight-forward expressf1 and f2 as

16

f1(Tk) ≤132xk+1

∑i=1

2i +15

∑j=1

2·

132xk+1

∑i=1

(2yk +2 j16

xk +2i)

+

132xk+1

∑i=1

(2xk +2yk +2i), (27)

and

f2(Tk) ≤132yk−1+1

∑i=1

2i +15

∑j=1

2·

132yk−1+1

∑i=1

(2xk−1 +2 j16

yk−1 +2i)

(28)

+

132yk−1+1

∑i=1

(2xk−1 +2yk−1 +2i), (29)

(30)

respectively. Further, plugging (27) and (28) into (26) andthen (26) into (25) we yieldthe recursion:

Φ(Tk) ≤ 1024·Φ(Tk−2)+20,323,353

984,06432k +o(32k). (31)

Hereby, we omit the value for the recursion startT2 because it is of no importance forthe coefficient of then· log2n term.

After resolving (31) and applying the result to (24) one gets

Φ(Tk) ≤60,970,0591,968,128

·32k ·k+o(32k ·k).

Finally, making use of the special dimension, i.e.

√n =

7831

·32k/2− 1631

,

one can state the following upper bound:

Φ(Tk) ≤6,774,4516,922,240

·n· log2n+o(n· log2n).

We summarize the section on upper bounds on large grids by stating the followinglemma:

Lemma 8.Let GN,N denote theN×N square planar grid withn = N2 vertices andwith N = 78

31 ·32k/2 + 1531 for some even integerk. Then the size of a minimum strictly

fundamental cycle bases onGN,N can be bounded from above by

0.978·n· log2 n+O(n).

Once again remember that hereby the previously best asymptotical upper bound byAlon et al. ([1]) is enhanced by a factor of more than four third.

17

4.2 Small Grids

Unfortunately, the 3 : 1−block structured trees, as described in the above section, arenot perfectly suited for smaller dimensions. This is due to the fact that, although asymp-totically 3 : 1 turned out to be a very good aspect ratio for theblocks, it is not possibleto decompose a square grid into such blocks without losing much of their advantagebecause of rounding “errors.”

Therefore, for small grids, we chose a different block-structured graph. This timewe demand a fix aspect ratio of 2 : 1. Moreover, the 2 : 1−blocks, in a sense, do notcover, but rather tile the square grid. The tiling procedureroughly goes as follows:

At first, two opposite 2 : 1−blocks are put in the middle of the grid. See for examplethe bold line bordered vertical blocks with side lengths 8× 15 in Figure 7. Then,horizontal 2 : 1−blocks are added centrally aside such that rectangular subgrids in thefour corners remain. Now in such a corner we always direct thenext block such thatits depth can be chosen as small as possible, always staying as close as possible tothe target ratio 2 : 1. During this procedure we do not pay attention to any roundinginaccuracies. In Figure 7 an example 2 : 1−block structured tree for dimensionN = 31is shown.

Figure 7: Notice the parquet-like structure of the tree withtiles having height-widthratio of 2 with small errors due to roundings. Inside, the blocks themselves are recur-sively filled with smaller blocks still maintaining the 2 : 1 ratio.

5 Experimental Results

In this section we compare different spanning trees with respect to the length of thestrictly fundamental cycle basis they induce. In addition to the degree-based tree-growing heuristics that we already referred to in the Introduction, local search tech-niques have also been considered. The most natural neighborhood that one can thinkof in the context of spanning trees, is the 2-neighborhood (cf. Section 2).

Amaldi et al. ([2]) reported the performance of several strategies for searching thistype of neighborhood. In what they denote by local search (LS), the entire neighbor-hood is examined and they move to the tree with the best improvement. In a sec-

18

ond deterministic strategy (ES), only those neighbors are tested, having a fixed ratioof branches according to a predefined order. To prevent LS to terminate too earlyin a too bad local optimum, Amaldi et al. ([2]) run metaheuristics such as variable-neighborhood search (VNS) and a tabu search (TS) on top of LS.In any of their com-putations, an adapted version of the tree-growing heuristic in [20] is used as the initialsolution.

In our computations, we use the 2 : 1−block-structured tree as initial solution.To improve them, in contrast to (LS) we do not examine the entire 2-neighborhood.Rather, whenever we identify a neighbor that improves the current solution, we greedilymove to that neighbor. Of course, this method depends on the order the edges in thetree are checked. Empirical studies showed, however, that the influence of the edge-order is neglectable. For our computational studies we chose a random order of edgesand ran our greedy-like approach—denoted by (GS)—ten times, considering best andaverage values of both length of the cycle basis and the running time of (GS).

Results. In Table 1 we compare the constructive heuristics, i.e. those that build upa tree without doing any subsequent local improvements. Moreover, we complementthese values with information on lower bounds, once obtained by Corollary 6 and thetrivial ones by a minimum weakly fundamental cycle basis (MWFCB). Notice that thelatter were also used in the recent study of Amaldi et al. ([2]). In addition to theselower bound values, let us mention that forN = 130—being the dimension closestto 100 for which our asymptotic bound is defined exactly—the bound that we derivedin Corollary 6 is only by about 3.5% weaker.

In our tables theitalic numbers highlight the best known upper and lower bounds.For N = 5, these coincide, and we mark this inboldface. Observe that for any dimen-sion N ≥ 10, the new trees that we propose in Section 4 yield smaller SFCB valuesthan any of the other constructive heuristics.

In Table 2 we compare the different local-search-type heuristics. For our greedysearch (GS) we used a 3.2GHz Intel P4 computer (“A1”), running Linux and usingLEDA c©. Amaldi et al. used for their local search heuristics (LS) and (ES) also an IntelP4 computer running Linux, but with 2.66GHz (“A2”). Accordingly, the times statedin Table 2 refer to the particular architecture. The values for the metaheuristics (TS)and (VNS)—also quoted from [2]—each refer to 10 minute runs on the A2 environ-ment. Much similar as in the purely constructive context, our new solutions improvethe best known upper bounds for all dimensionsN ≥ 20.

As already mentioned before we ran our local search (GS) witha random order. InTable 2 the first column presents the value and running time for the best run out of 10samples, and the deviations are very small. However, it has to be annotated that onlyfor dimensionsN∈ {60,80,90,100} the start tree hadnotalready been locally optimal.

6 Conclusions

Any serious summary of this paper has to remain a bit twofold.On the one hand, onsquare planar grid graphs—being a particularly challenging family of graphs for the

19

N “2 : 1′′ AKPW Machete C-Order Deo’s NTCorollary 6 MWFCB[1] [5] [16] [8, 16]

5 76 78 72 72 78 72 6410 468 524 492 492 518 422 32415 1 300 1 554 1 512 1 512 1 588 1 072 78420 2 550 3 030 3 382 3 382 3 636 2 022 1 44425 4 368 5 410 6 352 6 352 6 452 3 272 2 30430 6 656 8 408 10 672 10 672 11 638 4 822 3 36435 9 592 11 694 16 592 16 592 16 776 6 672 4 62440 13 162 16 078 24 362 24 362 28 100 8 822 6 08445 17 236 21 784 34 232 34 232 35 744 11 272 7 74450 21 920 27 912 46 452 46 452 48 254 14 022 9 60455 27 356 35 124 61 272 61 272 62 026 17 072 11 66460 33 406 42 790 78 942 78 942 92 978 20 422 13 92470 47 300 59 244 123 832 − − 28 022 19 04480 63 964 80 678 183 122 − − 36 822 24 96490 83 412 108 012 258 812 − − 46 822 31 684

100 106 090 137 390 352 902 − − 58 022 39 204

Table 1: Comparison of the cost of some selected trees, i.e. the length of the accord-ing strictly fundamental cycle bases. The rightmost columnpresents the previouslybest lower bound for small dimensions, obtained just by 4· (N−1)2. The penultimatecolumn now states the consistently better lower bounds due to Corollary 6.

MSFCB Problem—we we have improved significantly the lower and upper boundsthat were previously known for the MSFCB Problem.

On the other hand, just reconsider the rows withN = 10 in Tables 1 and 2. For thisrelatively small dimension, we simply believe the optimality gap that we are leavinghere (about 10%) should better not the that big. Even worse for N = 8: Indeed, inFigure 8 we provide a spanning tree with smaller SFCB value than the one on the firstpage—and which only everyeighthparticipant of MATHEON’s 2006 christmas quizhas been aware of. Yet, we are not aware of any concise combinatorial proof for itsoptimality. Hence, further efforts are to be made.

Nevertheless, columns 4–6 of Table 1 illustrate impressively to what extent degree-based heuristics for the MSFCB problem are inferior to applying recursive approaches.In other words, for any heuristic for the MSFCB problem whichwill be designed onlyin the future, we strongly recommend to evaluate it also on planar square grid graphs(in contrast to what has been done in some studies in the past,e.g. [7, 8]), and therecompare its performance to theitalic values that we provide in Tables 1 and 2.

20

N (GS) (LS) (ES) (VNS) (TS)cost time cost time cost time cost cost

5 72 0:00:00 72 0:00:00 74 0:00:00 72 7210 468 0:00:00 474 0:00:00 524 0:00:00 466 46615 1 300 0:00:00 1 318 0:00:00 1 430 0:00:00 1 280127620 2 550 0:00:00 2 608 0:00:03 3 186 0:00:00 2 572 259025 4 368 0:00:00 4 592 0:00:16 5 152 0:00:02 4 464 443030 6 656 0:00:01 6 956 0:00:47 8 488 0:00:03 6 900 688235 9 592 0:00:02 10 012 0:02:19 11 662 0:00:08 9 982 996440 13 162 0:00:07 13 548 0:06:34 15 924 0:00:26 13 524 1353445 17 236 0:00:06 18 100 0:14:22 22 602 0:01:00 18 100 1810050 21 920 0:00:09 23 026 0:31:04 33 274 0:01:10 23 026 2355260 33 374 0:01:01 − − − − − −80 63 810 0:07:24 − − − − − −90 83 222 0:07:48 − − − − − −

100 105 766 0:14:01 − − − − − −

Table 2: An overview of the quality of five local search approaches. Missing values aremarked with an “−” and running times are measured inh:mm:ss. The columns (LS)–(VNS) are cited from [2].

Acknowledgment

The authors would like to thank Janina Brenner for intensivediscussions on the empir-ical lower bounds.

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[19] C. Liebchen and G. Wunsch. The zoo of tree spanner problems. Technical Report2006-013, TU Berlin, Mathematical Institute, 2006.

[20] K. Paton. An algorithm for finding a fundamental set of cycles of a graph.Com-munications of the ACM, 12(9):514–518, 1969.

[21] R. Rizzi. Manuscript, 2006.

[22] N. Robertson and P. D. Seymour. Graph minors. V. Excluding a planar graph.J.Comb. Theory, Ser. B, 41(1):92–114, 1986.

[23] A. Schrijver. Combinatorial Optimization, volume 24 ofAlgorithms and Combi-natorics. Springer, 2003.

[24] H. Whitney. On the abstract properties of linear dependence.American Journalof Mathematics, 57:509–533, 1935.

23

Reports from the group

“Combinatorial Optimization and Graph Algorithms”

of the Department of Mathematics, TU Berlin

2006/32 Romeo Rizzi and Christian Liebchen:A New Bound on the Length of Minimum CycleBases

2006/24 Christian Liebchen and Sebastian Stiller:Delay Resistant Timetabling

2006/08 Nicole Megow and Tjark Vredeveld:Approximation Results for Preemptive StochasticOnline Scheduling

2006/07 Ekkehard Kohler and Christian Liebchen and Romeo Rizzi and Gregor Wunsch: Re-ducing the Optimality Gap of Strictly Fundamental Cycle Bases in Planar Grids

2006/05 Georg Baier and Thomas Erlebach and Alexander Hall and Ekkehard Kohler andHeiko Schilling:Length-Bounded Cuts and Flows

2005/30 Ronald Koch and Martin Skutella and Ines Spenke :Maximum k-Splittable Flows

2005/29 Ronald Koch and Ines Spenke :Complexity and Approximability of k-Splittable Flows

2005/28 Stefan Heinz and Sven O. Krumke and Nicole Megow and Jorg Rambau and AndreasTuchscherer and Tjark Vredeveld:The Online Target Date Assignment Problem

2005/18 Christian Liebchen and Romeo Rizzi:Classes of Cycle Bases

2005/11 Rolf H. Mohring and Heiko Schilling and Birk Schutz and Dorothea Wagner andThomas Willhalm:Partitioning Graphs to Speed Up Dijkstra’s Algorithm.

2005/07 Gabriele Di Stefano and Stefan Krause and Marco E. Lubbecke and Uwe T.Zimmermann:On Minimum Monotone and Unimodal Partitions of Permutations

2005/06 Christian Liebchen:A Cut-based Heuristic to Produce Almost Feasible Periodic Rail-way Timetables

2005/03 Nicole Megow, Marc Uetz, and Tjark Vredeveld:Models and Algorithms for StochasticOnline Scheduling

2004/37 Laura Heinrich-Litan and Marco E. Lubbecke:Rectangle Covers Revisited Computa-tionally

2004/35 Alex Hall and Heiko Schilling:Flows over Time: Towards a more Realistic and Com-putationally Tractable Model

2004/31 Christian Liebchen and Romeo Rizzi:A Greedy Approach to Compute a MinimumCycle Bases of a Directed Graph

2004/27 Ekkehard Kohler and Rolf H. Mohring and Gregor Wunsch:Minimizing Total Delayin Fixed-Time Controlled Traffic Networks

2004/26 Rolf H. Mohring and Ekkehard Kohler and Ewgenij Gawrilow and Bjorn Stenzel:Conflict-free Real-time AGV Routing

2004/21 Christian Liebchen and Mark Proksch and Frank H. Wagner:Performance of Algo-rithms for Periodic Timetable Optimization

2004/20 Christian Liebchen and Rolf H. Mohring: The Modeling Power of the Periodic EventScheduling Problem: Railway Timetables — and Beyond

2004/19 Ronald Koch and Ines Spenke:Complexity and Approximability of k-splittable flowproblems

2004/18 Nicole Megow, Marc Uetz, and Tjark Vredeveld:Stochastic Online Scheduling onParallel Machines

2004/09 Marco E. Lubbecke and Uwe T. Zimmermann:Shunting Minimal Rail Car Allocation

2004/08 Marco E. Lubbecke and Jacques Desrosiers:Selected Topics in Column Generation

2003/050 Berit Johannes: On the Complexity of Scheduling Unit-Time Jobs with OR-Precedence Constraints

2003/49 Christian Liebchen and Rolf H. Mohring: Information on MIPLIB’s timetab-instances

2003/48 Jacques Desrosiers and Marco E. Lubbecke:A Primer in Column Generation

2003/47 Thomas Erlebach, Vanessa Kaab, and Rolf H. Mohring: Scheduling AND/OR-Networks on Identical Parallel Machines

2003/43 Michael R. Bussieck, Thomas Lindner, and Marco E. Lubbecke:A Fast Algorithm forNear Cost Optimal Line Plans

2003/42 Marco E. Lubbecke:Dual Variable Based Fathoming in Dynamic Programs for Col-umn Generation

2003/37 Sandor P. Fekete, Marco E. Lubbecke, and Henk Meijer:Minimizing the StabbingNumber of Matchings, Trees, and Triangulations

2003/25 Daniel Villeneuve, Jacques Desrosiers, Marco E. Lubbecke, and Francois Soumis:OnCompact Formulations for Integer Programs Solved by ColumnGeneration

2003/24 Alex Hall, Katharina Langkau, and Martin Skutella:An FPTAS for Quickest Multi-commodity Flows with Inflow-Dependent Transit Times

2003/23 Sven O. Krumke, Nicole Megow, and Tjark Vredeveld:How to Whack Moles

2003/22 Nicole Megow and Andreas S. Schulz:Scheduling to Minimize Average CompletionTime Revisited: Deterministic On-Line Algorithms

2003/16 Christian Liebchen:Symmetry for Periodic Railway Timetables

2003/12 Christian Liebchen:Finding Short Integral Cycle Bases for Cyclic Timetabling

762/2002 Ekkehard Kohler and Katharina Langkau and Martin Skutella:Time-ExpandedGraphs for Flow-Dependent Transit Times

761/2002 Christian Liebchen and Leon Peeters:On Cyclic Timetabling and Cycles in Graphs

752/2002 Ekkehard Kohler and Rolf H. Mohring and Martin Skutella:Traffic Networks andFlows Over Time

739/2002 Georg Baier and Ekkehard Kohler and Martin Skutella:On thek-splittable FlowProblem

736/2002 Christian Liebchen and Rolf H. Mohring: A Case Study in Periodic Timetabling

Reports may be requested from: Sekretariat MA 6–1Fakultt II – Institut fr MathematikTU BerlinStraße des 17. Juni 136D-10623 Berlin – Germany

e-mail: klink@math.TU-Berlin.DE

Reports are also available in various formats from

http://www.math.tu-berlin.de/coga/publications/techreports/

and via anonymous ftp as

ftp://ftp.math.tu-berlin.de/pub/Preprints/combi/Report-number-year.ps